Recall that if $f(z)$ is analytic in a neighbourhood of $z_{0} \neq 0$, then

$$f(z)+\overline{f\left(z_{0}\right)}=2 u\left(\frac{z+\overline{z_{0}}}{2}, \frac{z-\overline{z_{0}}}{2 i}\right)$$

where $u(x, y)$ is the real part of $f(z)$. Use this fact to construct the imaginary part of an analytic function whose real part is given by

$$u(x, y)=y \int_{-\infty}^{\infty} \frac{g(t) d t}{(t-x)^{2}+y^{2}}, \quad x, y \in \mathbb{R}, y \neq 0$$

where $g(t)$ is real and has sufficient smoothness and decay.